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Let () be a polynomial equation, where P is a univariate polynomial of degree n. If one divides all coefficients of P by its leading coefficient, one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial. For example, the equation
Given a quadratic polynomial of the form + + it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial. Example: + + = [+ +] = [(+) +] = (+) + = (+) + This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms.
Let n be a positive integer, which will be the degree of the equation that we will consider, and (X 1, ..., X n) an ordered list of indeterminates.According to Vieta's formulas this defines the generic monic polynomial of degree n = + = = = (), where E i is the i th elementary symmetric polynomial.
The roots of the characteristic polynomial () are the eigenvalues of ().If there are n distinct eigenvalues , …,, then () is diagonalizable as () =, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ 's: = [], = [].
Graph of the polynomial function x 4 + x 3 – x 2 – 7x/4 – 1/2 (in green) together with the graph of its resolvent cubic R 4 (y) (in red). The roots of both polynomials are visible too. In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four:
MPSolve (Multiprecision Polynomial Solver) is a package for the approximation of the roots of a univariate polynomial. It uses the Aberth method, [1] combined with a careful use of multiprecision. [2] "Mpsolve takes advantage of sparsity, and has special hooks for polynomials that can be evaluated efficiently by straight-line programs" [3]
This equation immediately gives the k-th Newton identity in k variables. Since this is an identity of symmetric polynomials (homogeneous) of degree k, its validity for any number of variables follows from its validity for k variables. Concretely, the identities in n < k variables can be deduced by setting k − n variables to zero.
For example, finding a substitution = + + for a cubic equation of degree =, = + + + such that substituting = yields a new equation ′ = + ′ + ′ + ′ such that ′ =, ′ =, or both. More generally, it may be defined conveniently by means of field theory , as the transformation on minimal polynomials implied by a different choice of ...